Properties

Label 507.2.j.i
Level $507$
Weight $2$
Character orbit 507.j
Analytic conductor $4.048$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{8} + 2 \beta_{11} ) q^{2} + \beta_{7} q^{3} + ( 3 - \beta_{4} - 3 \beta_{7} - \beta_{9} ) q^{4} + ( \beta_{2} - 3 \beta_{6} + 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{5} + ( -\beta_{1} - 2 \beta_{2} ) q^{6} + ( \beta_{2} - \beta_{6} ) q^{7} + ( 3 \beta_{2} + 2 \beta_{6} + 3 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{8} + ( -1 + \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{8} + 2 \beta_{11} ) q^{2} + \beta_{7} q^{3} + ( 3 - \beta_{4} - 3 \beta_{7} - \beta_{9} ) q^{4} + ( \beta_{2} - 3 \beta_{6} + 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{5} + ( -\beta_{1} - 2 \beta_{2} ) q^{6} + ( \beta_{2} - \beta_{6} ) q^{7} + ( 3 \beta_{2} + 2 \beta_{6} + 3 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{8} + ( -1 + \beta_{7} ) q^{9} + ( 4 - 4 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - \beta_{7} + 4 \beta_{9} ) q^{10} + ( \beta_{1} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{11} + ( 4 - \beta_{3} ) q^{12} + ( -1 - 2 \beta_{3} ) q^{14} + ( -2 \beta_{1} + 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{15} + ( -5 + 5 \beta_{3} - 6 \beta_{4} - \beta_{5} - 5 \beta_{9} ) q^{16} + ( 1 + 3 \beta_{4} - \beta_{7} + \beta_{9} ) q^{17} + ( -2 \beta_{2} - \beta_{8} - 2 \beta_{11} ) q^{18} + ( \beta_{1} + \beta_{2} - 3 \beta_{6} ) q^{19} + ( 7 \beta_{1} + 2 \beta_{2} - 8 \beta_{6} ) q^{20} + ( \beta_{2} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{21} + ( 7 + 7 \beta_{4} - 7 \beta_{7} + 5 \beta_{9} ) q^{22} + ( -1 + \beta_{3} + 4 \beta_{4} + 5 \beta_{5} - \beta_{7} - \beta_{9} ) q^{23} + ( -3 \beta_{1} + 3 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{24} + ( -4 - 2 \beta_{3} + 3 \beta_{5} ) q^{25} - q^{27} + ( -\beta_{1} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{28} + ( -2 + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{7} - 2 \beta_{9} ) q^{29} + ( 1 - 2 \beta_{4} - \beta_{7} + 4 \beta_{9} ) q^{30} + ( -2 \beta_{2} + 5 \beta_{6} + 3 \beta_{8} - 5 \beta_{10} - 2 \beta_{11} ) q^{31} + ( 7 \beta_{1} + 5 \beta_{6} ) q^{32} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{6} ) q^{33} + ( -\beta_{2} - 7 \beta_{8} - \beta_{11} ) q^{34} + ( -3 - \beta_{4} + 3 \beta_{7} - 4 \beta_{9} ) q^{35} + ( 1 - \beta_{3} + \beta_{4} + 3 \beta_{7} + \beta_{9} ) q^{36} + ( -\beta_{1} + \beta_{8} + 8 \beta_{10} + \beta_{11} ) q^{37} + ( 3 - 5 \beta_{3} - 4 \beta_{5} ) q^{38} + ( 4 - \beta_{3} - 8 \beta_{5} ) q^{40} + ( -\beta_{1} + \beta_{8} - 4 \beta_{10} - 2 \beta_{11} ) q^{41} + ( 2 - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} + 2 \beta_{9} ) q^{42} + ( -3 + 4 \beta_{4} + 3 \beta_{7} + 2 \beta_{9} ) q^{43} + ( 3 \beta_{2} - 4 \beta_{6} - 7 \beta_{8} + 4 \beta_{10} + 3 \beta_{11} ) q^{44} + ( -2 \beta_{1} - \beta_{2} + 3 \beta_{6} ) q^{45} + ( -12 \beta_{1} - 2 \beta_{2} + 7 \beta_{6} ) q^{46} + ( -\beta_{2} + 3 \beta_{6} - \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{47} + ( -6 \beta_{4} - 5 \beta_{9} ) q^{48} + ( -2 + 2 \beta_{3} - \beta_{4} + \beta_{5} - 4 \beta_{7} - 2 \beta_{9} ) q^{49} + ( -7 \beta_{1} + 7 \beta_{8} - \beta_{10} - 7 \beta_{11} ) q^{50} + ( \beta_{3} - 2 \beta_{5} ) q^{51} + ( -8 + 4 \beta_{3} + 3 \beta_{5} ) q^{53} + ( \beta_{1} - \beta_{8} - 2 \beta_{11} ) q^{54} + ( 10 - 10 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} - 5 \beta_{7} + 10 \beta_{9} ) q^{55} + ( -1 - 3 \beta_{4} + \beta_{7} - \beta_{9} ) q^{56} + ( \beta_{2} - 3 \beta_{6} + \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{57} + ( -6 \beta_{1} - 3 \beta_{2} + 7 \beta_{6} ) q^{58} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{6} ) q^{59} + ( 2 \beta_{2} - 8 \beta_{6} + 7 \beta_{8} + 8 \beta_{10} + 2 \beta_{11} ) q^{60} + ( 3 + \beta_{4} - 3 \beta_{7} - 5 \beta_{9} ) q^{61} + ( -13 + 13 \beta_{3} - 16 \beta_{4} - 3 \beta_{5} + 9 \beta_{7} - 13 \beta_{9} ) q^{62} + ( \beta_{10} + \beta_{11} ) q^{63} + ( 7 \beta_{3} - 7 \beta_{5} ) q^{64} + ( 2 + 5 \beta_{3} - 2 \beta_{5} ) q^{66} + ( 4 \beta_{1} - 4 \beta_{8} - 4 \beta_{10} + 3 \beta_{11} ) q^{67} + ( 5 - 5 \beta_{3} + 14 \beta_{4} + 9 \beta_{5} - 2 \beta_{7} + 5 \beta_{9} ) q^{68} + ( 1 + 4 \beta_{4} - \beta_{7} - \beta_{9} ) q^{69} + ( -5 \beta_{2} + 11 \beta_{6} - 4 \beta_{8} - 11 \beta_{10} - 5 \beta_{11} ) q^{70} + ( -7 \beta_{1} - 2 \beta_{2} + 3 \beta_{6} ) q^{71} + ( -3 \beta_{1} - 3 \beta_{2} - 2 \beta_{6} ) q^{72} + ( -3 \beta_{2} + \beta_{6} + 6 \beta_{8} - \beta_{10} - 3 \beta_{11} ) q^{73} + ( 2 + 6 \beta_{4} - 2 \beta_{7} + 15 \beta_{9} ) q^{74} + ( 2 - 2 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} - 6 \beta_{7} + 2 \beta_{9} ) q^{75} + ( 2 \beta_{1} - 2 \beta_{8} - 8 \beta_{10} - \beta_{11} ) q^{76} + ( -3 - 4 \beta_{3} - 2 \beta_{5} ) q^{77} + ( 3 - 3 \beta_{3} - 3 \beta_{5} ) q^{79} + ( 5 \beta_{1} - 5 \beta_{8} + 6 \beta_{10} + 3 \beta_{11} ) q^{80} -\beta_{7} q^{81} + ( -7 - 9 \beta_{4} + 7 \beta_{7} - 9 \beta_{9} ) q^{82} + ( -\beta_{2} - 4 \beta_{6} + \beta_{8} + 4 \beta_{10} - \beta_{11} ) q^{83} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{6} ) q^{84} + ( \beta_{1} - 2 \beta_{2} - 6 \beta_{6} ) q^{85} + ( -10 \beta_{2} - 2 \beta_{6} - 13 \beta_{8} + 2 \beta_{10} - 10 \beta_{11} ) q^{86} + ( -1 + \beta_{4} + \beta_{7} - 2 \beta_{9} ) q^{87} + ( 5 - 5 \beta_{3} + 14 \beta_{4} + 9 \beta_{5} - 2 \beta_{7} + 5 \beta_{9} ) q^{88} + ( 6 \beta_{1} - 6 \beta_{8} - 4 \beta_{10} - 7 \beta_{11} ) q^{89} + ( -3 + 4 \beta_{3} + 6 \beta_{5} ) q^{90} + ( -4 + 19 \beta_{5} ) q^{92} + ( -3 \beta_{1} + 3 \beta_{8} - 5 \beta_{10} - 2 \beta_{11} ) q^{93} + ( -5 + 5 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{7} - 5 \beta_{9} ) q^{94} + ( -10 - 5 \beta_{4} + 10 \beta_{7} - 4 \beta_{9} ) q^{95} + ( 5 \beta_{6} + 7 \beta_{8} - 5 \beta_{10} ) q^{96} + ( 12 \beta_{1} + 10 \beta_{2} - 9 \beta_{6} ) q^{97} + ( 5 \beta_{1} + 9 \beta_{2} + 5 \beta_{6} ) q^{98} + ( -2 \beta_{2} + 2 \beta_{6} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{3} + 22q^{4} - 6q^{9} + O(q^{10}) \) \( 12q + 6q^{3} + 22q^{4} - 6q^{9} - 2q^{10} + 44q^{12} - 20q^{14} - 22q^{16} - 2q^{17} + 18q^{22} - 44q^{25} - 12q^{27} + 4q^{29} + 2q^{30} - 8q^{35} + 22q^{36} + 12q^{40} - 10q^{42} - 30q^{43} + 22q^{48} - 30q^{49} - 4q^{51} - 68q^{53} - 6q^{55} + 2q^{56} + 26q^{61} - 4q^{62} + 36q^{66} + 26q^{68} - 30q^{74} - 22q^{75} - 60q^{77} + 12q^{79} - 6q^{81} - 6q^{82} - 4q^{87} + 26q^{88} + 4q^{90} + 28q^{92} - 42q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -25 \nu^{11} + 95 \nu^{9} - 361 \nu^{7} + 155 \nu^{5} - 30 \nu^{3} - 1563 \nu \)\()/559\)
\(\beta_{3}\)\(=\)\((\)\( 25 \nu^{10} - 95 \nu^{8} + 361 \nu^{6} - 155 \nu^{4} + 30 \nu^{2} + 1004 \)\()/559\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{10} - 20 \nu^{8} + 76 \nu^{6} - 139 \nu^{4} + 124 \nu^{2} - 24 \)\()/43\)
\(\beta_{5}\)\(=\)\((\)\( 45 \nu^{10} - 171 \nu^{8} + 538 \nu^{6} - 279 \nu^{4} + 54 \nu^{2} + 242 \)\()/559\)
\(\beta_{6}\)\(=\)\((\)\( 70 \nu^{11} - 266 \nu^{9} + 899 \nu^{7} - 434 \nu^{5} + 84 \nu^{3} + 1246 \nu \)\()/559\)
\(\beta_{7}\)\(=\)\((\)\( 114 \nu^{10} - 545 \nu^{8} + 2071 \nu^{6} - 2831 \nu^{4} + 3379 \nu^{2} - 95 \)\()/559\)
\(\beta_{8}\)\(=\)\((\)\( 114 \nu^{11} - 545 \nu^{9} + 2071 \nu^{7} - 2831 \nu^{5} + 3379 \nu^{3} - 95 \nu \)\()/559\)
\(\beta_{9}\)\(=\)\((\)\( -128 \nu^{10} + 710 \nu^{8} - 2698 \nu^{6} + 4483 \nu^{4} - 4402 \nu^{2} + 852 \)\()/559\)
\(\beta_{10}\)\(=\)\((\)\( -242 \nu^{11} + 1255 \nu^{9} - 4769 \nu^{7} + 7314 \nu^{5} - 7781 \nu^{3} + 1506 \nu \)\()/559\)
\(\beta_{11}\)\(=\)\((\)\( -317 \nu^{11} + 1540 \nu^{9} - 5852 \nu^{7} + 8338 \nu^{5} - 9548 \nu^{3} + 1848 \nu \)\()/559\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{11} + 3 \beta_{8} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(3 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 2\)
\(\nu^{5}\)\(=\)\(4 \beta_{11} - \beta_{10} + 9 \beta_{8} - 9 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{5} + 9 \beta_{3} - 14\)
\(\nu^{7}\)\(=\)\(-5 \beta_{6} - 14 \beta_{2} - 28 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-28 \beta_{9} - 14 \beta_{7} - 19 \beta_{5} - 47 \beta_{4} + 28 \beta_{3} - 28\)
\(\nu^{9}\)\(=\)\(-47 \beta_{11} + 19 \beta_{10} - 89 \beta_{8} - 19 \beta_{6} - 47 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-89 \beta_{9} - 42 \beta_{7} - 155 \beta_{4} + 42\)
\(\nu^{11}\)\(=\)\(-155 \beta_{11} + 66 \beta_{10} - 286 \beta_{8} + 286 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(1 - \beta_{7}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
1.56052 0.900969i
−1.07992 + 0.623490i
−0.385418 + 0.222521i
0.385418 0.222521i
1.07992 0.623490i
−1.56052 + 0.900969i
1.56052 + 0.900969i
−1.07992 0.623490i
−0.385418 0.222521i
0.385418 + 0.222521i
1.07992 + 0.623490i
−1.56052 0.900969i
−2.33136 1.34601i 0.500000 0.866025i 2.62349 + 4.54402i 1.04892i −2.33136 + 1.34601i −0.480608 + 0.277479i 8.74094i −0.500000 0.866025i −1.41185 + 2.44540i
316.2 −2.04113 1.17845i 0.500000 0.866025i 1.77748 + 3.07868i 3.69202i −2.04113 + 1.17845i 0.694498 0.400969i 3.66487i −0.500000 0.866025i 4.35086 7.53590i
316.3 −1.77441 1.02446i 0.500000 0.866025i 1.09903 + 1.90358i 3.35690i −1.77441 + 1.02446i 1.94594 1.12349i 0.405813i −0.500000 0.866025i −3.43900 + 5.95652i
316.4 1.77441 + 1.02446i 0.500000 0.866025i 1.09903 + 1.90358i 3.35690i 1.77441 1.02446i −1.94594 + 1.12349i 0.405813i −0.500000 0.866025i −3.43900 + 5.95652i
316.5 2.04113 + 1.17845i 0.500000 0.866025i 1.77748 + 3.07868i 3.69202i 2.04113 1.17845i −0.694498 + 0.400969i 3.66487i −0.500000 0.866025i 4.35086 7.53590i
316.6 2.33136 + 1.34601i 0.500000 0.866025i 2.62349 + 4.54402i 1.04892i 2.33136 1.34601i 0.480608 0.277479i 8.74094i −0.500000 0.866025i −1.41185 + 2.44540i
361.1 −2.33136 + 1.34601i 0.500000 + 0.866025i 2.62349 4.54402i 1.04892i −2.33136 1.34601i −0.480608 0.277479i 8.74094i −0.500000 + 0.866025i −1.41185 2.44540i
361.2 −2.04113 + 1.17845i 0.500000 + 0.866025i 1.77748 3.07868i 3.69202i −2.04113 1.17845i 0.694498 + 0.400969i 3.66487i −0.500000 + 0.866025i 4.35086 + 7.53590i
361.3 −1.77441 + 1.02446i 0.500000 + 0.866025i 1.09903 1.90358i 3.35690i −1.77441 1.02446i 1.94594 + 1.12349i 0.405813i −0.500000 + 0.866025i −3.43900 5.95652i
361.4 1.77441 1.02446i 0.500000 + 0.866025i 1.09903 1.90358i 3.35690i 1.77441 + 1.02446i −1.94594 1.12349i 0.405813i −0.500000 + 0.866025i −3.43900 5.95652i
361.5 2.04113 1.17845i 0.500000 + 0.866025i 1.77748 3.07868i 3.69202i 2.04113 + 1.17845i −0.694498 0.400969i 3.66487i −0.500000 + 0.866025i 4.35086 + 7.53590i
361.6 2.33136 1.34601i 0.500000 + 0.866025i 2.62349 4.54402i 1.04892i 2.33136 + 1.34601i 0.480608 + 0.277479i 8.74094i −0.500000 + 0.866025i −1.41185 2.44540i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.i 12
13.b even 2 1 inner 507.2.j.i 12
13.c even 3 1 507.2.b.f 6
13.c even 3 1 inner 507.2.j.i 12
13.d odd 4 1 507.2.e.i 6
13.d odd 4 1 507.2.e.l 6
13.e even 6 1 507.2.b.f 6
13.e even 6 1 inner 507.2.j.i 12
13.f odd 12 1 507.2.a.i 3
13.f odd 12 1 507.2.a.l yes 3
13.f odd 12 1 507.2.e.i 6
13.f odd 12 1 507.2.e.l 6
39.h odd 6 1 1521.2.b.k 6
39.i odd 6 1 1521.2.b.k 6
39.k even 12 1 1521.2.a.n 3
39.k even 12 1 1521.2.a.s 3
52.l even 12 1 8112.2.a.cg 3
52.l even 12 1 8112.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.f odd 12 1
507.2.a.l yes 3 13.f odd 12 1
507.2.b.f 6 13.c even 3 1
507.2.b.f 6 13.e even 6 1
507.2.e.i 6 13.d odd 4 1
507.2.e.i 6 13.f odd 12 1
507.2.e.l 6 13.d odd 4 1
507.2.e.l 6 13.f odd 12 1
507.2.j.i 12 1.a even 1 1 trivial
507.2.j.i 12 13.b even 2 1 inner
507.2.j.i 12 13.c even 3 1 inner
507.2.j.i 12 13.e even 6 1 inner
1521.2.a.n 3 39.k even 12 1
1521.2.a.s 3 39.k even 12 1
1521.2.b.k 6 39.h odd 6 1
1521.2.b.k 6 39.i odd 6 1
8112.2.a.cg 3 52.l even 12 1
8112.2.a.cp 3 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{12} - 17 T_{2}^{10} + 195 T_{2}^{8} - 1260 T_{2}^{6} + 5963 T_{2}^{4} - 15886 T_{2}^{2} + 28561 \)
\( T_{5}^{6} + 26 T_{5}^{4} + 181 T_{5}^{2} + 169 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 28561 - 15886 T^{2} + 5963 T^{4} - 1260 T^{6} + 195 T^{8} - 17 T^{10} + T^{12} \)
$3$ \( ( 1 - T + T^{2} )^{6} \)
$5$ \( ( 169 + 181 T^{2} + 26 T^{4} + T^{6} )^{2} \)
$7$ \( 1 - 5 T^{2} + 19 T^{4} - 28 T^{6} + 31 T^{8} - 6 T^{10} + T^{12} \)
$11$ \( 2825761 - 796794 T^{2} + 155755 T^{4} - 16072 T^{6} + 1207 T^{8} - 41 T^{10} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( ( 169 - 208 T + 243 T^{2} - 42 T^{3} + 17 T^{4} + T^{5} + T^{6} )^{2} \)
$19$ \( 2401 - 4802 T^{2} + 8575 T^{4} - 1960 T^{6} + 343 T^{8} - 21 T^{10} + T^{12} \)
$23$ \( ( 8281 - 4459 T + 2401 T^{2} - 182 T^{3} + 49 T^{4} + T^{6} )^{2} \)
$29$ \( ( 841 - 435 T + 283 T^{2} - 28 T^{3} + 19 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$31$ \( ( 38809 + 7985 T^{2} + 174 T^{4} + T^{6} )^{2} \)
$37$ \( 20200652641 - 1235527397 T^{2} + 51974835 T^{4} - 1158780 T^{6} + 18863 T^{8} - 166 T^{10} + T^{12} \)
$41$ \( 707281 - 1020974 T^{2} + 1412403 T^{4} - 86940 T^{6} + 4115 T^{8} - 73 T^{10} + T^{12} \)
$43$ \( ( 1681 + 1927 T + 1594 T^{2} + 623 T^{3} + 178 T^{4} + 15 T^{5} + T^{6} )^{2} \)
$47$ \( ( 49 + 98 T^{2} + 21 T^{4} + T^{6} )^{2} \)
$53$ \( ( -41 + 66 T + 17 T^{2} + T^{3} )^{4} \)
$59$ \( 116985856 - 16267264 T^{2} + 1526528 T^{4} - 80640 T^{6} + 3120 T^{8} - 68 T^{10} + T^{12} \)
$61$ \( ( 27889 - 2672 T + 2427 T^{2} - 126 T^{3} + 185 T^{4} - 13 T^{5} + T^{6} )^{2} \)
$67$ \( 2825761 - 2040734 T^{2} + 1115743 T^{4} - 255220 T^{6} + 44155 T^{8} - 213 T^{10} + T^{12} \)
$71$ \( 1698181681 - 341251729 T^{2} + 61074923 T^{4} - 1424724 T^{6} + 24843 T^{8} - 182 T^{10} + T^{12} \)
$73$ \( ( 851929 + 29301 T^{2} + 306 T^{4} + T^{6} )^{2} \)
$79$ \( ( 27 - 18 T - 3 T^{2} + T^{3} )^{4} \)
$83$ \( ( 1849 + 649 T^{2} + 62 T^{4} + T^{6} )^{2} \)
$89$ \( 163047361 - 130575794 T^{2} + 102004507 T^{4} - 2029888 T^{6} + 30175 T^{8} - 201 T^{10} + T^{12} \)
$97$ \( 7181161893361 - 255465058539 T^{2} + 7514975158 T^{4} - 50599759 T^{6} + 249238 T^{8} - 587 T^{10} + T^{12} \)
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